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Chapter – II MF radar and MF radar and MF radar and MF radar and
Data Analysis TechniquesData Analysis TechniquesData Analysis TechniquesData Analysis Techniques
Chapter II
53
Chapter II
MF radar and Data Analysis Technique
2.1 Introduction
In this thesis, horizontal wind measurements collected by MF radars from
Kolhapur (16.8°N, 74.2°E, 10.6° N dip lat.) and Tirunelveli (8.7°N, 77.8°E, 0.3°
N dip lat) form the basic data source. The main objectives of collecting data
from these radars are to study and understand the dynamics and coupling
processes of winds in MLT region at low latitudes.
This chapter can be conveniently divided into two sections. The first
section of this chapter is devoted to the details of MF radar systems installed at
Kolhapur and Tirunelveli, mode of operation and the method of wind estimation,
beginning with the basics of scattering mechanisms. Both the radars have been
installed and operated by the Indian Institute of Geomagnetism. They have been
providing nearly continuous wind information of the mesosphere and lower
thermosphere region using spaced antenna method. The radar technique utilizes
partial reflection from weakly ionized atmospheric layers. A full correlation
analysis (FCA) is used to extract wind information from the data [1] and a brief
account on this is given in this chapter.
The interpretation of radar echoes from the higher heights sampled by the
radar, in terms of neutral winds is complicated, as the measured drifts at highest
(above 92 km) probed by the radar are influenced by the electrodynamical
processes unique to the equatorial region. There is another problem due to total
reflection, as there will be a difference in the true and virtual heights as the
transmitted electromagnetic ray approaches the total reflection heights, which
normally occurs above 94 km. A brief discussion on these aspects is presented in
this chapter.
Chapter II
54
The second section of this chapter describes a few mathematical
techniques namely standard Fourier technique, time-domain filtering and
singular and bispectral analysis used for examining the MF radar data.
2.2 Observational Techniques
Many different techniques are currently used to study the atmosphere
including radars [2], lidars [3], Satellites [4], Balloons [5] and Rockets [6], each
with its own advantages and disadvantages. Ground based remote sensing
techniques such as radars (RAdio Detection And Ranging) and Lidars (LIght
Detection And Ranging) have the advantages of continuous measurements with
good time and height resolution.
The instruments we use for the measuring the earth’s atmosphere decides
much of our knowledge of the state of the earth’s atmosphere. The parameters of
primary interest are temperature, pressure, humidity, precipitation and wind.
Since the nineteenth century, the routine surface measurements of these
parameters have been made through out the world. We began to get a systematic
look at the space and time variability of the atmosphere as a function of height
above the ground in the last century. With the development of the rocket and
radiosonde, a significant advance in this area was made. These devices carried
by a balloon upwards, measures temperature, pressure and humidity and transmit
these data back to a ground station. Also the winds could be determined by
tracking the motion of the balloon. Rocket observations are categorized
according to the pay load of the measuring system. Among them the falling
sphere and datasonde have often been launched because these payloads are small
and less expensive [7]. Their observational height ranges from 30-90 km and 20-
65 km respectively. Rocket and radiosonde observations are limited to 2-4
soundings per day, so that these observations are only appropriate for defining
the large scale planetary waves [8].
Advances in our understanding of the atmosphere are dependent upon the
measurements with a higher resolution in both time and space. Many
Chapter II
55
applications require more frequent and more closely-spaced upper air
measurements. Though the radiosondes are inexpensive, the relative high cost of
maintaining balloon launching and tracking facilities has prevented more
extensive use. Development of remote sensing technology offers a solution to
this problem which gives more successive data in height and time using radio
waves and optical rays.
The temperature and wind perturbations of the global scale atmospheric
waves can be observed in selected height region by choosing specific frequency
bands. The instruments on board the Upper Atmospheric Research Satellite such
as WIND imaging interferometer (WINDII) and High Resolution Doppler
Imager (HRDI) can estimates atmospheric winds and temperatures on a global
scale. Satellite based instruments generally have good altitude resolution and
excellent spatial coverage and hence can obtain global data by orbiting around
the planet. Since WINDII and HRDI have an effective resolution of a few
hundred kilometers, they do not readily detect gravity waves of short period, but
are more suitable for planetary waves and tides [9, 10]. These instruments have
to be validated by comparisons with ground based instruments like radar, lidar
etc. The first form of the ground based radar was ionosonde which sweeps the
entire high frequency (HF) region of the electromagnetic spectrum. The total
reflection occurs for the ionospheric electron densities in the HF region and the
ionosondes obtain reflections from different altitudes at different frequencies, as
atmospheric electron density changes. The ionograms are then used to determine
the vertical profile of electron densities.
MST radars can detect weak backscattering arising by refractive index
fluctuations in the mesosphere, stratosphere and lower thermosphere in the very
high frequency (VHF and UHF) region [11]. The MST radar technique was
developed from the incoherent radar scatter, which is used to obtain the weak
partial reflections due to incoherent scatter from free electrons in the ionosphere.
The MST radar technique is capable of obtaining strong and useful echo returns
over the height range of 1-100 km. Figure 2.1 illustrates a good example of
relative echo powers for various MST radar facilities [12]. As can be seen from
Chapter II
56
the figure below ~ 50 km, echoes arise primarily from the fluctuations in the
refractive index due to small scale turbulence. Water vapour becomes the most
important factor in the lower troposphere and atmospheric density dominates up
to the stratopause. In the mesosphere, the echoes arise from the neutral
turbulence fluctuations which are enhanced by free electrons. At heights above ~
80 km, HF and VHF radars can obtain reflections from incoming meteors and
hence the MST radar operating in this mode is called Meteor radar. When the
meteors traverse the atmosphere, intense ionization columns are formed at the
heights of the mesosphere and lower thermosphere. These ionized meteor trails
scatter radio waves, before they diffuse. Observations of meteors allow
estimation of neutral wind velocities by their bulk motion.
The MF radars can be used to study the partial reflections in the D region.
In this case the radio waves are coherently backscattered from refractive index
variations caused by a mixture of turbulence and wave motion. As the ionization
is strongly coupled to the neutral air, measurements of scatter motions can be
associated with motion of the neutrals. These neutral winds can be estimated and
in this thesis the neutral winds are collected by using such partial reflections.
Chapter II
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Figure 2.1: Relative echo power profiles for various MST radar facilities
[after Gage and Balsley (1984)] [12]
Radars, including MF and meteor systems, have been widely used for the
determination of mean winds, tides and the creation of seasonal and global
climatology. Both types of systems provided mean winds for CIRA 1986 and
results were inherently consistent in magnitude and directions for heights of 80-
100 km at many latitudes. Comparison with the satellite data in CIRA 1986
demonstrates good agreement below 80 km, but there was inadequacy in the
satellite data above 80 km [13]. This has been addressed by Hedin et al. (1996)
[14] in the semi-empirical horizontal wind model (HWM-93). Regarding tides, a
Chapter II
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summary of climatology from meteor and MF radars with model comparisons
was the focus of the special issue of the Journal of Atmospheric and Terrestrial
Physics in July/August 1989. Such observations [15, 16] demonstrated good
agreement between radar types at similar latitudes. Models, particularly of the
semidiurnal tide [17] were in good agreement at mid-latitudes. Various radar
measurements of the middle atmosphere have clarified the characteristics of
planetary wave [18, 19] and gravity waves [2, 20]. For these reasons the MF
radar measurements being used by the community have been highly regarded
and widely used.
Several different techniques are applied in radar measurements of
atmospheric winds in the middle atmosphere. The two ground based techniques
which can directly measure the neutral winds on the continuous basis include the
Doppler method [21] and the spaced antenna method [22, 1, 23 and 24]. Here,
first the Doppler method will be briefly described and then some time is spent on
describing the spaced antenna method as this technique was applied to the MF
radars for collecting the wind data in the present work.
2.3 Radar scattering and reflection mechanisms
Radio waves can be backscattered from the atmosphere due to variations
of the refractive index of the air [25]. The refractive index depends on the
amount of bound electrons as well as the amount of free electrons in the air. The
free electrons are produced along with positive ions by the ionization of a
number of atmospheric constituents (including molecular oxygen, molecular
nitrogen and nitrogen monoxide) by high energy solar radiation and cosmic rays.
The ionosphere is the region of the atmosphere that contains sufficient
electron densities that the propagation of radio waves will be affected. A
common way of classifying the ionosphere is by the peaks in the vertical profile
of electron density. This results in regions, with ascending height termed the D,
E, F1 and F2 regions [26]. The D region corresponds approximately with the
mesosphere and lies between about 60-90 km. The ionization in this region is
Chapter II
59
formed mostly by the ionization of nitric oxide by the solar Lyman-α hydrogen
emission line.
The concentration of electrons varies spatially throughout the ionosphere
depending on height, season, time of day and other factors such as the amount of
solar activity. During the polar winter, when solar ionization is absent, electron
precipitation allows radio echoes to occur. The electron concentrations are
significantly higher during the day than at night because of the absence of solar
radiation at night combined with the recombination of the electrons and positive
ions.
The ionosonde was the first form of radar used for atmospheric research.
Ionosondes operate by sweeping through a range of frequencies in the high
frequency (HF) range of the electromagnetic spectrum. Since different
frequencies reflect from different altitudes as the atmospheric electron density
changes, a vertical profile of electron densities, known as an ionogram, can be
obtained.
Medium frequency (MF) radars operate in the frequency range 1-3 MHz.
Radio waves of this frequency are coherently backscattered from sharp gradients
in the refractive index caused by turbulence and small-scale wave motion or thin
sheets of laminar flow in the D-region of the ionosphere.
Above about 50 km, solar radiation leads to an increase in free electron
density with height. Bellow about 100 km there is a high collision frequency
between the ionized particles and the neutral atmosphere causing the ionized
particles to be strongly coupled to the motion of the neutral atmosphere. This
allows the neutral wind to be estimated from the partial reflections of the MF
radio waves, and is the essence of how MF radars can measure wind speeds.
There are other varities of atmospheric radars which operates in different
frequency ranges. Radars that operate in the very high frequency (VHF) range
and the ultra high frequency (UHF) range derive their measurements from weak
backscattered signals arising from refractive index fluctuations [11]. Radars of
this variety are commonly known as MST radars because they have height
coverage in the mesosphere, stratosphere and troposphere.
Chapter II
60
2.3.1 Refractive index
Gage and Balsley [1980] [25] has given a formula on refractive index for
radio waves propagating through the lower and middle atmosphere as
n-1= 3.73 × 10-1 e/T2 + 77.6 × 10-6 P/T - Ne/2Nc (2.1)
where e is the partial pressure of water vapour in mb, T is the absolute
temperature, P is the atmospheric pressure in mb, Ne is the number density of
free electrons and Nc is the critical plasma density. The first term is the
contribution due to water vapour, and is dominant in the troposphere due to the
presence of relatively large amount of water vapour. The second term is due to
dry air and thus dominates above the tropopause. The third term represents
neutral turbulence induced electron density fluctuations and is dominant at
ionospheric heights (above 50 km).
The scattering/reflection mechanisms responsible for the radar signal
return from the atmosphere is broadly classified into coherent scatter and
incoherent scatter.
Coherent scatter results from macroscopic fluctuations in refractive index
associated with clear air turbulence. In this type of scatter, the echo power
depends on the density gradients associated with the coherent scatters. There are
three types of coherent scatter, namely, Bragg scatter, Fresnel reflection and
Fresnel scatter that are depicted in figure 2.2. We will briefly consider each of
these in turn.
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61
Figure 2.2: Artist’s conception of atmospheric refractivity structure
pertinent to Bragg scatter (A), Fresnel scatter (B), and Fresnel reflection
(C) [27].
1. Bragg scatter
Bragg scatter or turbulent scatter is generally a coherent scattering
phenomenon that results from variations in the refractive index with a scale
projected along the line of sight of the radar equal to one-half the radar
wavelength. Bragg scatter can be isotropic i.e. without causing a radar aspect
sensitivity. The term aspect sensitivity is typically used to describe the decrease
of radar return signal power with increasing beam pointing zenith angle; there
can also be a smaller variation of radar return signal power as a function of beam
Chapter II
62
azimuth angle. Small aspect sensitivity means that the scatterers are specular
with the backscattered power falling rapidly with zenith angle [28]. This occurs
when the turbulent irregularities of refractive index are randomly distributed in
the vertical direction but exhibit coherency in the horizontal direction. Bragg
scatter can be anisotropic, causing the aspect sensitivity, if the statistical
properties of the irregularities, namely their correlation distances, are dependent
on direction. Although the aspect sensitivity is different for these two processes,
the temporal variations of the radar echoes should be similar because of the
randomly fluctuating irregularities. Their Doppler spectrum approximately
reveals a Gaussian shape.
2. Fresnel scatter
Fresnel scattering occurs from many closely spaced layers randomly
distributed in height. In the case Bragg scatter, the scattering scale in the
horizontal are less than the size of the first Fresnel zone and many scatterers are
involved. Therefore, the scatter produces tends to be vary randomly in amplitude
and phase and is relatively weak. Fresnel scatter produces much stronger and
more coherent scattered signals.
3. Fresnel reflection
Fresnel reflection occurs from a sharp vertical gradient in the refractive
index that is horizontally coherent over a scale greater that a Fresnel zone. It is
same as that of the Fresnel scattering model except that the backscattered signals
are from a single extended layer of sheet. In this case the returned signal may be
strong and will be quite steady in amplitude and phase over relatively long
periods. Fresnel reflections are responsible for “slow fading” of radio signals
backscattered from the mesosphere at medium frequencies. Radar returns from
the mesosphere, particularly at long wavelengths, are generally refered to as
partial reflections. However, there is observational evidence to suggest that the
Chapter II
63
radar returns are due to both Fresnel reflections from coherent electron density
gradients and scattering from turbulent irregularities.
Incoherent scatter is also known as Thomason scatter, or thermal scatter.
This type of scatter occurs when the incident electromagnetic field interacts
directly with free electrons, which are forced to oscillate by the electric field
component. These oscillating electrons re-radiate at a frequency similar to the
incident radiation. If the wavelength of the incident radiation is small enough,
then the reradiated signal is phase-incoherent. Since the back scattered radar
signal arises from small electron density fluctuations due to random thermal
motions of the ions and electrons, it has to rely on the Thomson scattering cross
section of the electron and hence it is extremely weak. As a consequence of this,
incoherent scatter radars must possess relatively large power-aperture products
in order to observe these backscattered signals.
2.4 Doppler method
The Doppler method uses a large (often steerable) antenna to form a
narrow beam, and then directs a large fraction of the available transmitter power
along this “pencil beam”. The radio waves scatter from irregularities within the
beam, and are scattered to a receiving array which collects the signal. In many
cases the receiver array is the same as the transmitter array and this arrangement
is called monostatic radar. Figure 2.3 demonstrates schematically this technique.
The returning signal is Doppler shifted relative to those transmitted, due to the
motion of the scatterer, and the radar then measures this Doppler shift. Typical
values for the Doppler shift are in range 0.01-10 Hz. The value of the Doppler
shift is then is used to determine the radial velocity of the scatterer within the
beam, and by using combinations of the radial velocities obtained with the
different beam pointing directions, it is possible to make measurements of the
nett motion of the atmosphere. For a constant zenith angle and uniform wind
field, the radial velocities vary sinusoidally with the azimuth angle. The Fourier
analysis of the radial velocities yields the eastward wind and the northward wind
from the Fourier coefficients.
Chapter II
64
Figure 2.3: Schematic diagram demonstrating the main principles of the
Doppler method. The Doppler shift of the returning radiation relative to
that transmitted is indicated, and this shift is the basis of this method [29].
2.5 Spaced Antenna method
Earlier, a crude method was designed to calculate the wind vector from
time series sampled from spaced antennae [22]. This method was known as
“method of similar fades”. Mitra [1949] [22] used this technique to measure the
Chapter II
65
ionospheric drifts from the amplitude fadings of the echoes reflected from the
ionosphere. This method is based on the assumption that the features of the
fading recorded at the three antennas are similar but displaced in time with
respect to each other. From the time delays, one can easily determine the
velocity of the fading patterns over ground, which is twice the velocity of the
irregularities of the ionosphere. However, the pattern may changes as it moves
and add to the signal variation and the method described above is not applicable.
Briggs et al. [1950] [30] developed a method called ‘full correlation
analysis’ (FCA) that take into account the random change in the diffraction
pattern. The analysis calculated the auto- and cross-correlation functions for the
three fading records, obtained from the three receiving antennas, at different
time lags to extract the background wind vector. This wind vector obtained
without considering random changes in the pattern is called apparent velocity
and is, in general, an over-estimate of the background wind velocity. It is
corrected for random changes in the diffraction pattern to obtain true velocity.
Originally developed in 1950’s, the space antenna (SA) method, used in
conjunction with MF radars, has proved to be an important and relatively
inexpensive technique for making measurements of atmospheric wind velocities
and other parameters. Phillips and Spenser [1955] [31] extended the analysis to
anisotropic case to find the size and shape of the pattern as well as the velocity
parameters corrected for the anisotropy effects .It was originally used for total
reflection experiments, subsequently, modified for D-region work using partial
reflections [32, 33] and still later used for tropospheric and stratospheric
measurements with VHF radars [34, 35]. The SA method generally utilizes
different antenna for transmission and reception and determines the drift
velocities of the ionised irregularities that partially reflects the radar signal in the
D-region depending on time and season. Measurements are limited to the middle
atmosphere and lower thermosphere where weakly ionized turbulent structures
can be expected to move with neutral motions due to the high collision
frequency. Many discussions of the full correlation analysis in its current form
Chapter II
66
and some of the computational difficulties involved in its implementation are
available in the literature [36, 1, 23].
In this analysis, a moving pattern over spatially arranged three sensors at
the vortices of an equilateral triangle is generally represented by a family of
ellipsoids of the forms
ρ(ξ, η, τ) = ρ(A(ξ- Vx τ)2 + B(η- Vy τ)2 + 2H(ξ- Vx τ)( η- Vy τ)) + K τ2 (2.2)
ρ(ξ, η, τ) = ρ(Aξ2 + Bη2 + Cτ2 + 2Fξη + 2Gητ +2Hξη) (2.3)
where ρ(ξ, η, τ) is the cross-correlation function, which is a function of ξ,
η, spatial separation between any two receivers in x and y-directions respectively
and τ, the time shift with the two receivers. Vx and Vy are the component
velocities of the moving pattern with respect to a stationary observer. The
constants A, B, C, F, G and H fully describe the size, shape of the pattern and the
velocities. Hence the evaluation of these constants provides the velocity and
anisotropy parameters.
Maximum correlation occurs when ∂ρ/∂τ = 0 and the corresponding time
shift
τ = -(F/C)ξ - (G/C)η (2.4)
It is assumed that the cross-correlation between two receivers at zero time shift
is equal to the auto-correlation function of the receivers. Mathematically,
ρ(0, 0, τ) = ρ(ξ, η, 0) (2.5)
Applying the above equation in equation (2.2), we find
ρ(Cτ2) = ρ(Aξ2 + Bη2 + 2Hξη), (2.6)
and thus
τ2 = (A/C)ξ2 + (B/C)η2 + (2H/C)ξη (2.7)
Chapter II
67
The above equation (2.7) is for a pair of sensors and it gives the time shift at
which the mean auto correlation function (averaged over all sensors) has the
value equal to the value of cross-correlation between the pair of sensors at zero
time shift. Likewise, we get three equations for the three pair of sensors. The
coefficients A/C, B/C and H/C can be estimated simultaneously and will be in
general be over-estimated and by using least squares technique the optimum
values and errors can be determined.
By equating the coefficients of ξη and ητ in equations (2.1) and (2.2), we obtain,
AVx + HVy = - F, BVy + HVx = - G. (2.8)
These equations can be solved simultaneously to obtain the components of true
velocity Vx and Vy. Then the magnitude and direction of the true velocity are
given by
|V|2 = Vx2 + Vy
2 , (2.9)
tanφ = Vx/Vy (2.10)
The spatial properties of the pattern itself can be determined using spatial
correlation function,
ρ(ξ, η, 0) = ρ(Aξ2 + Bη2 + 2Hξ η), (2.11)
They are usually specified by giving the values of the minor axis of the
characteristics ellipse (A particular ellipse for which, ρ(ξ, η, 0) = 0.5), the axial
ratio, and the orientation of the major axis, measured clockwise from north.
The orientation θ of the major axis is given by
tan2θ = 2H/(B-A), clockwise from the y-axis (2.12)
Chapter II
68
The major and minor axes are respectively,
a = (2Cτ0.52) / [(A+B) - (√ (4H2 + (A-B)2)] (2.13)
b = (2Cτ0.52) / [(A+B) + (√ (4H2 + (A-B)2)] (2.14)
The axial ratio r is given by the ratio of the major to the minor axes.
The random changes of the moving pattern are described by a parameter called
‘mean lifetime’ or ‘time-scale’. This is the mean lifetime of the irregularities
which cause the diffraction pattern to change randomly and is often taken to be
the time lag at which the auto-correlation function falls to 0.5. Mathematically,
T0.5 2 = τ0.5
2 C/K (2.15)
where τ0.5 is the time lag for which the directly observed auto-correlation
function for a fixed observer falls to 0.5. To find the ratio C/K, the coefficient of
τ2 in equation (2.2) and (2.3) can be equated to obtain
C = AVx2 + BVy
2 + K + 2HVxVy (2.16)
Since Vx and Vy, and the ratios A/C, B/C, H/C, the above equation (2.16)
can be used to find C/K.
Some data are rejected, when the assumptions of the analysis are not valid
[1]. The following criteria are found satisfactory for the spaced antenna wind
analysis.
1. The mean signal is either very weak or so strong that the receivers are
saturated most of the time.
2. The fading is very shallow (signal almost constant). The standard
deviation should be at least 2% of the mean signal.
3. The signal to noise ratio is less than 6 dB.
4. The mean auto-correlation function has not fallen to at least 0.5 for the
maximum number of lags, which have been computed.
Chapter II
69
5. The cross-correlation functions have no maxima within the number of
lags which have been computed.
6. The cross-correlation functions are oscillatory, so that it is impossible to
identify the correct maxima.
7. The sum of the time displacements τ’ij does not sum to zero, as it ideally
should, for the data taken in pairs around the triangle of antenna. This
condition may be relaxed somewhat to give a more practical criterion: if
|∑ τ’ij |/∑| τ’ij |>0.2 the results are rejected. This quantity has been called
the “normalized time discrepancy”.
8. The computed value of Vc2 is negative, so that Vc is imaginary. This
criterion should not be applied too rigidly, because if the computed Vc2 is
only slightly negative, this probably means that the true value is zero, and
the small negative value has arisen from statistical fluctuations. Such data
may therefore be particularly good, indicating pure drift with negligible
random changes.
9. The computed coefficients indicate hyperbolic rather than elliptical
contours, i.e. data will be rejected, if H2 < AB.
10. The polynomial interpolation procedures break down at any stage.
11. It is advisable to reject any results for which the correlations of the full
correlation analysis are very large, i.e. those for which the apparent or
true velocities differ greatly in magnitude and or direction. Suitable
criteria are: the data will be rejected if 0.5 Vt ≥ Va ≥ 3Vt or |φt - φa | ≥ 400.
2.6 Checking the Polarization
Here brief description is given about which kind of polarization is
to be used for the radar transmission and reception. Since for this work
we have used data from two radar sites, one being equatorial site and
another from non-equatorial, it is important to know about the
polarization method used for the transmission and reception of the signal.
Chapter II
70
In normal running O (Ordinary) mode transmission and reception
will be used during the day and E (Extraordinary) mode transmission and
reception will be used during the night.
When the radar system is first installed it is necessary first to
empirically determine O and E mode transmission and then to empirically
determine O and E mode reception. The procedure which follows must be
performed during daytime. During the day, at heights above 70 km,
absorption of the transmitted signal is greatest for the E mode of
propagation. This is caused by the D-region of the ionosphere extending
to lower heights during the day. The received signals must be examined
carefully to check which polarization is being transmitted and received by
the equipment. If necessary, the polarization controls for transmission
and/or reception must be adjusted.
2.6.1 Non-Equatorial Sites
Non-equatorial sites use a circularly-polarised system. For
transmission, each pair of parallel antenna elements is driven in-phase. A
900 phase delay is introduced to the signal driving the orthogonal pair of
transmitter antenna elements. To change the sense of the transmitted
circularly-polarised signal, the 900 phase delay is applied to the other pair
of parallel antenna elements by means of a relay.
In reception, two orthogonal antenna elements are attached to each
receiver. Once again, a 900 phase delay is introduced to the signal from
one antenna. To change reception mode, a model switch relay is used to
switch the phase delay from one antenna element to the other.
2.6.2 Equatorial Sites
Equatorial sites use a linealy-polarised system. During
transmission only one parallel pair of antenna elements is used. To
Chapter II
71
change transmission mode, a model switch relay switches transmission
from one pair of parallel antennas to the other.
During reception only antenna of each of the orthogonal antenna
pairs feed each receiver. To change reception mode, a mode switch relay
is used to switch from one antenna to the other.
2.7 About the radar
Both the radar systems are similar to the system installed at
Christmas Island [37]. The schematic diagram of the MF radar system is
shown in figure 2.4. Both the radar systems have been installed by Indian
Institute of Geomagnetism .The MF radar at Tirunelveli has been installed
in the year 1992 and MF radar at Kolhapur have been installed in the year
1999. It consists of three sections.
2.7.1 The transmitting and receiving antennas
The system details, mode of operations and method of wind
estimation for both radar systems are the same as that installed at
Christmas Island [37]. The transmitting antenna array is arranged in a
square, and consists of four centre-fed half-wave dipoles, approximately
75 m in length. As discussed in above section, for the transmission and
reception of the signal, radar system at Tirunelveli being an equatorial site
use linear polarization while radar system at Kolhapur which is non-
equatorial site use circular polarization. The O-mode dipoles are used for
daytime transmission while the E-mode dipoles are used at nighttime.
This geometrical arrangement also applies to the three receiving antennas.
The ideal height of the transmitting antennas is one quarter of the
transmitting wavelength above the ground plane, which for MF radar
operating at 1.98 MHz is about 30 m.
The receiving antennas are of the inverted-V type, and are situated
at the vertices of an equilateral triangle whose basic spacing is 180 m.
Chapter II
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The centroids of both the transmitting and receiving arrays coincide,
making the radar system a monostatic. The impedance of each receiving
antenna is 50 Ω to match the coaxial cable that takes the signal to the
receiving system.
Figure 2.4: Schematic diagram of the MF radar system at Tirunelveli
showing the geometry of the transmitting and receiving antennas.
2.7.2 The transmitter, receiving and data acquisition system, and
computer controller
The transmitter, the receiving and data acquisition system (RDAS)
and the computer are the main components of the radar system. The
transmitting system uses totally solid-state transmitters. It consists of a
Rx
2 Rx1
Rx
3
Tx
1
75 m
CONTROL ROOM
180mm
Chapter II
73
combiner unit, 10 power amplifier modules, a fan unit, a driver unit and a
power supply unit. The transmitter operates at a frequency of 1.98 MHz
and is fully phase coherent. The transmitter power is 25 kW, with a pulse
length of 30 µs which corresponds to a 4.5 km height resolution. The
pulse repetition frequency is 80 Hz with 32 point coherent integration
during daytime. This is reduced to 40 Hz during nighttime with 16 point
integration, to avoid problems arising from multiple reflections from
equatorial spread F. Table 2.1 lists some of the characteristics of the
transmitting system.
The specifications of RDAS operating at Tirunelveli are listed in
table 2.2. The receiving system is controlled by a pc-based
microprocessor, which both controls the transmitter and acquires data.
The data are subsequently transferred to a host personal computer for
analysis and subsequent storage. The 1.98 MHz transmitter pulses are
derived from the master oscillator and frequency synthesis module. The
transmitted signal is linearly polarized. The three receivers are of
superheterodyne type. In each receiver, the received 1.98 MHz signal is
mixed with a 2.475 MHz local oscillator to produce a 495 kHz
intermediate frequency (IF), which is then fed to the signal processor
modules. The signal processors are phase sensitive: the 495 kHz IF
signals are heterodyned with in-phase and quadrature local oscillators to
produce in-phase and quadrature components, which are then digitized to
8-bit resolution.
Chapter II
74
Table 2.1: Characteristics of transmitting system of MF radar
Peak Transmit power 25 kW RMS
Maximum duty cycle 0.25 %
Operating frequency 1.98 MHz
Half power pulse width 30 µs
Pulse rise and fall times 15 µs
Output impedance 50 Ω unbalanced 30 kHz
Half power bandwidth 30 kHz
Harmonics
(third, fifth and even harmonics)
-63 dBc, >-70dBc, >-70dBc
Load An array of four 75 Ω dipoles
Overall efficiency ~ 70 %
Polarisation Linear/ circular
Height of transmitter towers 30 m
Table 2.2: Characteristics of receiving system of MF radar
No. of receivers 3
Sampling starting height 60-98 km (Day), 70-98 km (Night)
Sampling interval 2 km
Pulse repetition frequency 80 Hz (Day), 40 Hz (Night)
Coherent integrations 32 (Day), 16 (Night)
Samples per data set 256
Sampling period 2 minutes
Centre frequency 1.98 MHz
Bandwidth ~ 39 kHz
Sensitivity ~ 0.3 µV
Maximum gain ~ 105
Gain control Programmable in steps of 1 dB
Gain control range 60 dB
Input impedance 50 Ω
Chapter II
75
The RDAS acquires and stores in its memory a complete data set. A data
set comprises 256 points of 20 consecutive height samples at 2 km intervals, thus
providing a 40 km range coverage. Though the radar samples at every 2 km, the
radar pulse length of about 30 µs means, however, that the actual height of
resolution is about 4-5 km. Each data sample is the result of integrating the
digitized data over a number consecutive transmitter pulses. The transmitter
pulses are coherently averaged to produce a mean data point for every 0.4
seconds. A complete data set is therefore acquired in 102.4 seconds. It is then
transferred to the host computer. The coherent integration is done to improve
signal-to-noise ratio, which improves the performance of the system particularly
during nighttime, when the ionization is low. The other radar system details are
given in the tables 2.1 and 2.2.
The recording information is programmable by the host computer. It
includes the number of heights per sample, transmitter pulse repetition
frequency, integrations (Tx pulses) per sample point, sample points per data set
and receiver gain. Different recording configurations are used for daytime and
nighttime. The receiver gains are dynamically adjusted by the analysis program
before the start of accumulation of each data set.
2.7.3 Computer and analysis software
After each data acquisition run, the computer performs a full correlation
analysis on the 256 point complex data set to determine mean winds and various
other parameters, namely, pattern decay time, pattern axial ratio, signal-to-noise
ratio and the received power antenna. The system runs continuously for all the
days. However the data acquisition is interrupted while taking data backup and
during power failure. The data backup is normally done for every fortnight.
2.8 Data Analysis Techniques
The remainder of this chapter relies on obtaining meaningful and
statistical significant information of coherent oscillations present in the time
Chapter II
76
series of the radar wind measurements. To accomplish this, a background study
of mathematical techniques coupled with basic derivations, necessary for
understanding and investigating the time series, is presented. The results from
this analysis provides information on the probability of the existence of coherent
oscillations in a given data sample. This information is then used in interpreting
the source, properties and behaviour of these oscillations.
2.8.1 Harmonic Analysis
This method uses harmonic analysis, which is generally applied to a
discrete time series. Usually a function containing a constant term plus a number
of sinusoidal periodic terms at the frequencies where the main variance is
thought to be contained is fitted to that data using least square method, which is
described bellow.
In general, it is possible to fit a continuous physical process y(t) with a
function of the form (James, 1995),
T N= (2.17)
More formally:
1 1
1
( ) [ co s(2 ) sin (2 )], 0,1, 2 ...k K
n n k n k n
k
y t y a kT t b kT t kπ π=
− −
=
= = + =∑ (2.18)
where, ( )ny t = ny represents the discrete time samples of ( )y t .
By least square methods, the general best fit (for any of the k coefficients) is
given when < ε2 > is a minimum [38], where
2 1 1 2
1
[ c o s ( 2 ) s in ( 2 )]n N
n k n k n
n
y a kT t b kT tε π π=
− −
=
= − −∑ (2.19)
and
2 2
0k ka b
ε ε∂ ∂= =
∂ ∂ (2.20)
Chapter II
77
The solution is straightforward, the results being:
0 1 1 2 2 3 3( ) cos( ) sin( ) cos(2 ) sin(2 ) cos(3 ) sin(3 )f t a a t b t a t b t a t b tω ω ω ω ω ω= + + + + + + (2.21)
1 1 1
1 1
2 1 2 1
1 1
sin(2 ) sin(2 )cos(2 )
sin (2 ) sin (2 )
n N n N
n n n n
n n
n N n N
n n
n n
y kT t kT t kT t
bk ak
kT t kT t
π π π
π π
= =− − −
= =
= =− −
= =
= −∑ ∑
∑ ∑ (2.22)
When the length of the function ( )y t is an integer multiple of the period
/T k , for k = 1,2,3….., and the nt are equally spaced, i.e. when nt t n= ∆ × where
1t TM −∆ =
M = 1,2,…, 2
N, n = 1,2,…, N , and T N= , then the orthogonality rules apply,
and the equations 2.21 and 2.22 become
1
1 11
2 1 1
1
cos(2 )2 cos(2 )
cos (2 )
n N
n n n Nn
k n nn Nn
n
n
y kT t
a T y kT t
kT t
π
π
π
=−
=− −=
=− =
=
= =∑
∑∑
(2.23)
1
1 11
2 1 1
1
sin(2 )2 sin(2 )
sin (2 )
n N
n n n Nn
k n nn Nn
n
n
y kT t
b T y kT t
kT t
π
π
π
=−
=− −=
=− =
=
= =∑
∑∑
(2.24)
Note however, that 0a and 0b are special cases, where
01
1n N
n
n
a T y=
=
= − ∑ ; 0b =0. (2.25)
Chapter II
78
The least square fits were made for a constant wind 0a and solar tidal term with
periods of 48, 24, 12, hours according to the equation
0 1 1 2 2 3 3( ) cos( ) sin( ) cos(2 ) sin(2 ) cos(3 ) sin(3 )f t a a t b t a t b t a t b tω ω ω ω ω ω= + + + + + +
(2.26)
with k = 1, 2, 3 and 2
2 fT
πω π= = where f = 1 cycle per day (cpd).
2.8.2 Spectral analysis
A physical process can be described either in the time domain or in the
frequency domain. One representation can be written into other by means of
Fourier transform.
The Fourier transform may be written in discrete form as
exp(2 / ), 0 1.k jC c ijk N k toNπ= = −∑ KKK (2.27)
The discrete Fourier transform maps N complex numbers (the jc ’s) into N
complex numbers (the kc ’s). The computation of DFT described above involves
N2 complex computations. It can be computed in N log2 N , which is much less
than N 2 operations with an algorithm called the fast Fourier transform [39].
The Fourier coefficients obtained by using FFT/DFT are less then used to
estimate the power spectrum, which is the energy content of the signal at
particular frequency. The power spectrum curve shows how the variance of the
process is distributed with frequency. The variance contributed by frequencies in
the range f to f fδ+ is given by the area under the power spectrum curve
between the two ordinates f and f fδ+ . The periodogram estimate of the power
spectrum is given by
2 2 2( ) 1/ , 1,2,...,( / 2 1)k k N kP f N C C k N−= + = −KKK (2.28)
Where k
kf
N=
∆
Chapter II
79
The power spectral density (PSD) is given by 2 2( ) ( ) ( ) k k N kP k N P f N C C −= ∆ = ∆ + (2.29)
Amplitude 2 2( ) ( ) ( ) k k N kR P k N C C −= = ∆ + (2.30)
To test significance of the amplitude ( )kR , Nowroozi [1967] [40] has given a
method in which the probability P that the ratio 2 2k kR R∑ (summation runs
from k =1 to m) exceeds a parameter ‘g’ is given by
1 1 1 1(1 ) ( 1)(1 2 ) / 2 ........... ( 1) !(1 ) /( !( )!)m m m mP m g m m g m Lg L m L− − − −= − − − − + + − − − (2.31)
It can be shown that 2 22 /(2 1) ( ) k i meanR m X X= + −∑ ∑ .
It is, therefore, not necessary to calculate all the harmonics. The error introduced
in neglecting the higher order term is only 0.1% for p=0.05 (95% confidence
level). Therefore, the parameter g can be calculated only from P=m(1-g)m-1. The
value of g for different values of p and m is determined. The parameter gk is
given by
2 2(2 / ) ( ) k k i meang R N X X= −∑ (2.32)
If 0.05k pg g => (for 95% confidence level), the amplitude is 95% significant.
2.8.3 Digital filtering
To extract the signal lying only in a certain frequency band, the band-pass
filter is used. Filtering is more convenient in the frequency domain. The whole
data record is subjected to FFT. The FFT output is multiplied by a filter
function ( )H f . Inverse FFT is taken to get the filtered data set in time domain.
Chapter II
80
The nonrecursive or finite impulse response (FIR) filter is given by
0
( ) ex p ( 2 )M
k
k
H f c ik fπ=
= −∑ (2.33)
Here, the filter response function is just a discrete Fourier transform. The
transform is easily invertible, giving the desired small number of kc coefficients
in terms of the same small number of values of ( )iH f at some discrete
frequencies if . However, this fact is not very useful, as ( )H f will tend to
oscillate wildly between the discrete frequencies where it is pinned down to
specific values. Hence, the following procedure of [41] is adopted.
I. A relatively large value of M is chosen.
II. The M coefficients kc , k=0,…, M-1 can be found by an FFT.
III. Most of the kc ’s are set to zero except only the first 0 1 1( , , , )kk c c c −K and
last 1, 11( , )M K MK c c− − −− K .
IV. As the last few kc ’s are filer coefficients at negative lag, because of the
wraparound property of the FFT, the array of kc ’s are cyclically shifted to
bring everything to positive lag. The coefficients will be in the following
order 1 1 0 1 1( , , , , , , , 0 , 0 , 0 )M K M Kc c c c c− + − −K K K (2.34)
V. The FFT of the array will give an approximation to the original ( )H f . If
the new filter function is acceptable, then we will have a set of 2K-1 filter
coefficients. If it is not acceptable, either K is increased and the same
procedure is repeated or the magnitudes of the unacceptable ( )H f are
modified to bring it more in line with the original ( )H f , and then FFT is
taken to get new kc ’s.
VI. Now, all coefficients, except the first 2K-1 values are set to zero. Inverse
transform is taken to get a new ( )H f .
Chapter II
81
Estimation of the characteristics of planetary-scale waves
Unlike atmospheric tides, planetary-scale waves do not have fixed period.
They exhibit a range of periods and these periods may vary with height as well
as time. To compute the characteristics of these waves (the amplitude, phase and
the period), the following procedure is adopted. The hourly averaged wind data
are subjected to the time-domain filtering with finite impulse response (FIR)
filters [41] to get filtered data consisting of the particular planetary wave periods
only. The filtered data are then subjected to harmonic analysis with period varied
in the range of expected periods of concerned planetary wave. The wave
parameters are determined in least square sense.
2.8.4 Bispectral analysis
Power spectrum gives only the power of each spectral component and the
information about the phase relation between different spectral components is
suppressed. But, the bispectrum which is a higher order spectrum, is an
ensemble average of the product of three spectral components and can be used to
determine whether there is any phase relation between the three spectral
components. If each spectral component is independent of other components,
then the amplitude and phase of the component will be different from those of
other components and the bispectrum will give a nearly zero value. If the phase
associated with the three components sum to the same constant each time they
occur, then the bispectrum will give a non-zero value [42].
In the present analysis, the monthly data (720 points) are divided into ‘K’
segments containing ‘M’ samples (256 data points). 75% overlap between the
segments is ensured to get a large value for K. The mean of each segment is
calculated and removed. The DFT of each segment is computed. The bispectrum
of t hi segment is given by
1 2 1 2 1 2( , ) ( ) ( ) ( )i i i iB f f X f X f X f f= + (2.35)
Chapter II
82
where iX is the Fourier Transform of the thi segment.
The bispectral estimates are averaged across all segments. Then, they are
normalized by dividing each value by the maximum value, resulting in a relative
amplitude ranging from zero to one.
The bispectrum graphs are generally plotted with smaller frequencies
along y-axis and next higher frequencies along x-axis. It will only be non-zero at
locations 1 2( , )f f where 3 1 2f f f= ± and 3 1 2φ φ φ= ± , where iφ denotes the phase of
the thi component. Nonlinear interactions not only yield sum and difference
frequencies, but also phase relations, which are of the same form as frequency
relations. Such a phase relation is termed as quadratic phase coupling. The non-
zero value of bispectrum is the result of quadratic phase coupling indicating non-
linear interaction. If 3φ is random and independent of 1φ and 2φ , then the
bispectrum will be zero revealing that the phases are not related.
The interpretation of peaks in bispectral plots is given in [43]. As a simple
case, the presence of two frequencies and their sum denotes existence of
quadratic phase coupling (i.e., non-linear mixing). If the bispectrum shows large
value (peak) at any x-y point, it can be interpreted as a response to quadratic
phase coupling between x-frequency, y-frequency and the sum frequency. The x-
and y-coordinated (bifrequencies) account for two of the frequencies and the
sum frequency is represented by a diagonal through the peak, which intersects x-
and y-axes at the same value. The initial two mixing components could be at any
two of the three frequencies.
2.8.5 Singular spectral analysis
The power spectrum obtained using FFT gives only an average
contribution from a specific oscillation to the total variance and the phase
information is lost in deriving the power spectral density. Singular spectral
analysis extracts principal components of the variability even when the system is
non-stationary. This method generates data adaptive filters, whose transfer
functions highlight regions where sharp spectral peaks occur and thus helps
Chapter II
83
reconstruction of the original time series with just a few principal components
close to the spectral peaks. In contrast to the Fourier components, the principal
components need not be sinusoidal in nature. Briefly SSA decomposes the
original time series into its significant signal components with least noise [44].
Assume a finite time series ( )y t of length N.
( ) ( ), 1, 2,3 ,sy t y Kt K N= = KK (2.36)
and st is sampling interval. The series is normalized using the mean (Y) and
standard deviation ( )yσ . The new series will be
( ) ( ( ) ) / , 1, 2,3 ,yx t y t y t Nσ= − = KK (2.37)
The sampled time series is then embedded in an M-dimensional space, the
consecutive sequences of ( )X t
1 2
2 3 1
1
M
M
N M N
X X X
X X X
Z
X X
+
− +
=
K
K
K
K
K
(2.38)
The matrix so derived is called the Trajectory matrix. For different
choices of M, different trajectory matrices can be obtained. However, M should
be larger than the autocorrelation time (the lag at which the first zero occurs).
The eigen values of this matrix are then evaluated in descending order of
magnitude together with the corresponding eigen vectors. As the matrix is
positive symmetric Toeplitz (whose all diagonal elements are equal), the eigen
values will always be positive. Ideally, the number of non-zero eigen values will
correspond to the number of independent variables in the system. When quasi-
periodic fluctuations are present in the time series, the eigen vectors appear as
Chapter II
84
even/odd pair in phase quadrature, with corresponding eigen values nearly in
magnitude. As the PCs are filteres versions of the original series with the M
elements of the eigen vectors serving as appropriate filter weights, the resulting
series would be of length (N-M+1). Vautard et al. [1992] [45] has given a
method to extract a series of length N corresponding to a given set of eigen
elements which have been called the reconstructed components (RC). The
formulae for the thk component
( ) (1/ ) ,k k k
i i j jR X i a E−= ∑ for 1 ( 1)i M≤ ≤ − and j = 1 to M
= (1/ ) k k
i j jM a E−∑ , for ( 1)M i N M≤ ≤ − + and j = 1 to M (2.39)
= 1/( 1 1) k k
i j jN a E−− + ∑ , for ( 2)N M i N− + ≤ ≤ and j = 1 to M
where Ejk are the M eigen elements of the kth component and
k
ik i j jA X E+=∑ 1 ( )i N M≤ ≤ − and j = 1 to M (2.40)
The percentage of the total variance accounted by each of the
reconstructed component can be computed from the ratio of the individual eigen
value to the sum of all the M eigen values which then gives rise to an immediate
idea of the relative importance of a particular component to the time series.
2.8.6 Auto-and-Cross-correlation functions
The correlation between two continuous functions g(t) and h(t), which is
denoted by corr(g,h), and is a function of lag t. The correlation will be large at
some value of t if the first function g(t) is a close copy of the second h(t) but lags
in time by t, i.e., the first function is shifted to the right of the second. Likewise,
the correlation will be large for some negative value of the first function leads
Chapter II
85
the second, i.e., is shifted to the left of the second. The relation that holds when
the two functions are interchanged is
( , )( ) ( , )( )Corr g h t Corr h g t= − (2.41)
The discrete correlation of the two sampled functions gk and hk, each periodic
with time N, is defined by
( , ) j j k kC o r r g h g h+= ∑ (2.42)
According to the discrete correlation theorem, this discrete correlation of two
real functions g and h is one member of the discrete Fourier transform pair
( , ) *k kCorr g h G H⇔ (2.43)
where kG and kH are the discrete Fourier transforms of gj and hj+, and the
asterisk denotes complex conjugate.
The practical procedure adopted to find the correlation of two data sets
using FFT is given below.
I. FFT is taken for the two data sets
II. One resulting transform is multiplied by the complex conjugate of the
other.
III. Inverse transform is taken for the product
If the input data are real, the result (rk) will normally be a real vector of
length N. The components rk are the correlation at different lags (both positive
and negative). The correlation at zero lag is in r0, the first component; the
correlation at lag 1 is in r1, the second component; the correlation at lag -1 is in
rN-1, the last component; etc
Chapter II
86
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